To find the exact value of [tex]\(\sin \frac{2\pi}{3}\)[/tex], let's proceed step-by-step and utilize our knowledge of the unit circle and trigonometric identities.
1. Understanding the Angle [tex]\(\frac{2\pi}{3}\)[/tex]:
- The angle [tex]\(\frac{2\pi}{3}\)[/tex] is given in radians. To understand this better, we can convert it to degrees.
- [tex]\(\frac{2\pi}{3}\)[/tex] radians can be converted to degrees by multiplying by [tex]\(\frac{180^\circ}{\pi}\)[/tex]:
[tex]\[
\frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ
\][/tex]
- So, [tex]\(\frac{2\pi}{3}\)[/tex] radians is equivalent to 120 degrees.
2. Considering the Unit Circle:
- On the unit circle, 120 degrees is located in the second quadrant.
- In the unit circle, the sine of an angle is represented by the y-coordinate of the corresponding point.
3. Using the Reference Angle:
- The reference angle for 120 degrees is 180 degrees - 120 degrees = 60 degrees.
- Therefore, to find [tex]\(\sin 120^\circ\)[/tex], we can use the fact that [tex]\(\sin 120^\circ\)[/tex] is equal to [tex]\(\sin 60^\circ\)[/tex] because both angles are in the range of the sine function's symmetry in the unit circle.
4. Known Value of [tex]\(\sin 60^\circ\)[/tex]:
- From trigonometry, we know that [tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex].
5. Determining the Sign:
- Since 120 degrees lies in the second quadrant where the sine function is positive, the value of [tex]\(\sin 120^\circ = \sin 60^\circ\)[/tex] is positive.
Putting it all together, we have:
[tex]\[
\sin \frac{2\pi}{3} = \sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}
\][/tex]
Therefore, the exact value of [tex]\(\sin \frac{2\pi}{3}\)[/tex] is [tex]\(\boxed{\frac{\sqrt{3}}{2}}\)[/tex].
